For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

15. log  x15 y13 _ z19  16. ln  a −2 _

b−4 c5  17. log( √— x3 y−4 ) 18. ln  y √

_____

y _ 1 − y  19. log(x 2 y 3

3 √ —

x2 y5 )

For the following exercises, condense each expression to a single logarithm using the properties of logarithms. 20. log(2×4) + log(3×5) 21. ln(6×9) − ln(3×2) 22. 2log(x) + 3log(x + 1)

23. log(x) − 1 _ 2 log(y) + 3log(z) 24. 4log7 (c) +

log7(a) _ 3 + log7(b) _ 3

For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base. 25. log7(15) to base e 26. log14(55.875) to base 10

For the following exercises, suppose log5 (6) = a and log5 (11) = b. Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of a and b. Show the steps for solving.

27. log11(5) 28. log6(55) 29. log11  6 _ 11  nUmeRIC For the following exercises, use properties of logarithms to evaluate without using a calculator.

30. log3  1 _ 9  − 3log3 (3) 31. 6log8(2) + log8(64) _ 3log8(4)

32. 2log9(3) − 4log9(3) + log9  1 _ 729  For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.